LTL over Description Logic Axioms

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LTL over Description Logic Axioms

Franz Baader^?, Silvio Ghilardi, and Carsten Lutz

1 TU Dresden, Germany,baader@inf.tu-dresden.de

2 Universit`a degli Studi di Milano, Italy,ghilardi@dsi.unimi.it

3 TU Dresden, Germany,lutz@inf.tu-dresden.de

1 Introduction

In many applications of Description Logics (DLs) [7], such as the use of DLs as ontology languages or conceptual modeling languages, being able to represent dynamic aspects of the application domain would be quite useful. This is, for instance, the case if one wants to use DLs as conceptual modeling languages for temporal databases [4]. Another example are medical ontologies, where the faithful representation of concepts would often require the description of temporal patterns. As a simple example, consider the concept “Concussion with no loss of consciousness,” which is modeled as a primitive (i.e., not further defined) concept in the medical ontology SNOMED CT.¹ As argued in [18], a correct representation of this concept should actually say that, after the concussion, the patient remained conscious until the examination.

Since the expressiveness of pure DLs is not sufficient to describe such temporal patterns, a plethora of temporal extensions of DLs have been investigated in the literature.² These include approaches as diverse as the combination of DLs with Halpern and Shoham’s logic of time intervals [17], formalisms inspired by action logics [1], the treatment of time points and intervals as a concrete domains [13], and the combination of standard DLs with standard (propositional) temporal logics into logics with a two-dimensional semantics, where one dimension is for time and the other for the DL domain [15, 19, 11]. In this paper, we follow the last approach, where we use the basic DL ALC [16] in the DL component and linear temporal logic (LTL) [14] (sometimes also called propositional temporal logic (PTL) [11]) in the temporal component. However, even after the DL and the temporal logic to be combined have been fixed, there remain several degrees of freedom when defining the resulting temporalized DL.

On the one hand, one must decide to which pieces of syntax temporal operators can be applied. Temporal operators may be allowed to be use as concept constructors, as required by the above example of a concussion with no loss of consciousness, which could be defined using the until-operator U of LTL as follows:

∃finding.Concussionu Conscious U∃procedure.Examination. (1)

?Supported by NICTA, Canberra Research Lab.

1 see http://www.ihtsdo.org/our-standards/

2 For a more thorough survey of the literature on temporalized DLs, see the technical report accompanying this paper [8] and the survey papers [2, 3, 12].

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Alternatively or in addition, temporal operators may be applied to TBox axioms (i.e., general concept inclusions, GCIs) and/or to ABox assertions. For example, the temporalized TBox axiom

32(UScitizenv ∃insured by.HealthInsurer)

says that there is a future time point from which on US citizens will always have health insurance, and the formulaΨ:

3 (∃finding.Concussion)(BOB) ∧ (2)

Conscious(BOB)U(∃procedure.Examination)(BOB)

says that, sometime in the future, Bob will have a concussion with no loss of consciousness between the concussion and the examination.

On the other hand, one must decide whether one wants to have rigid concepts and/or roles, i.e., concepts/roles whose interpretation does not vary over time.

For example, the conceptHumanand the rolehas fathershould probably be rigid since a human being will stay a human being and have the same father over his/her life-time, whereasConsciousshould be a flexible concept (i.e., not rigid) since someone that is conscious at the moment need not always by conscious.

Similarly, insured by should be modeled as a flexible role. Using a logic that cannot enforce rigidity of concepts/roles may result in unintended models, and thus prevent certain useful inferences to be drawn. For example, the concept description∃has father.Humanu3(∀has father.¬Human) is only unsatisfiable if bothhas fatherandHumanare rigid.

The combination of (extensions of)ALCand LTL in which temporal operators can be applied to concept descriptions, TBox axioms, and ABox assertions has been studied by Wolter, Zakharyaschev, and others (see, e.g., [19, 11]). In particular, it is known that the basic reasoning problems are ExpSpace-complete.

In this setting, rigid concepts can be defined, but rigid roles cannot. In fact, as also shown in [11], the addition of rigid roles causes undecidability even w.r.t. a global TBox (i.e., where the same TBox axioms must hold at all time points).

Decidability can be regained by dropping TBoxes altogether, but the decision problem is still hard for non-elementary time. Decidable combinations of DLs and temporal logics that allow for rigid roles can be obtained by strongly restricting either the temporal component [6] or the DL component [5].

In this paper, we follow a different approach for regaining decidability in the presence of rigid roles: temporal operators are allowed to occur only in front of axioms (i.e., ABox assertions and TBox axioms), but not inside concept descriptions. We show that reasoning becomes simpler in this setting: with rigid roles, satisfiability is decidable (more precisely: 2-ExpTime-complete); without rigid roles, the complexity decreases to NExpTime-complete; and without any rigid symbols, it decreases further toExpTime-complete (i.e., the same complexity as reasoning in ALC alone). We also consider another way of decreasing the complexity of satisfiability to ExpTime: satisfiability without rigid roles (but with rigid concepts) becomes ExpTime-complete if GCIs can occur only as global

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axioms that must hold in every temporal world. Note that, in this case, ABox assertions arenot assumed to be global, i.e., the valid ABox assertions may vary over time.

The situation we concentrate on in this paper (i.e., where temporal operators are allowed to occur only in front of axioms) has been considered before only for the case where there are no rigid concepts or roles. The combination approach introduced in [10] yields a decision procedure for this case, whose worst-case complexity is, however, non-optimal. Our ExpTime upper bound for this case actually also follows from more general results in [11] (see the remark following Theorem 14.15 on page 605 of [11]). However, also in [11], the setting where temporal operators are allowed to occur only in front of axioms is considered only in the absence of rigid symbols.

Obviously, the temporalized DLs we investigate in this paper cannot be used to define temporal concepts such as (1) for concussion with no loss of consciousness. However, they are nevertheless useful in ontology-based applications since they can be used to reason about a temporal sequence of ABoxes w.r.t. a (global or temporalized) TBox. For example, in an emergency ward, the vital param- eters of a patient are monitored in short intervals (sometimes not longer than 10 minutes), and additional information is available from the patient record and added by doctors and nurses. Using concepts defined in a medical ontology like SNOMED CT, a high-level view of the medical status of the patient at a given time point can be given by an ABox. Obviously, the sequence of ABoxes obtained this way can be described using temporalized ABox assertions. Critical situations, which require the intervention of a doctor, can then be described by a formula in our temporalized DL, and recognized using the reasoning procedures developed in this paper. For example, given a formulaφencoding a sequence of ABoxes describing the medical status of Bob, starting at some time pointt0, and the formulaψ defined in (2), we can check whether Bob sometime aftert0 had a concussion with no loss of consciousness by testingφ∧ ¬ψfor unsatisfiability.

2 Basic definitions

We assume that the reader is familiar with the basic DL ALC [16] and with the temporal logic LTL [14]. We consider general ALC-TBoxes, i.e., TBoxes consist of finitely many general concept inclusion axioms (GCIs) of the form CvD, whereC, DareALC-concept descriptions. An ABox consists of a finite set of assertions of the form a : C or (a, b) : r where C is an ALC-concept description,ris a role name, anda, bare individual names. We call both GCIs and ABox assertions ALC-axioms. A Boolean combination of ALC-axioms is called aBooleanALC-knowledge base. For LTL, we use the variant with anon- strict until (U) and a next (X) operator. We are now ready to define our new logic, calledALC-LTL, whereALC-axioms replace propositional letters in LTL.

Definition 1. ALC-LTL formulae are defined by induction:

– ifαis anALC-axiom, then αis anALC-LTL formula;

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– ifφ, ψ areALC-LTL formulae, then so areφ∧ψ,φ∨ψ,¬φ,φUψ, andXφ.

As usual, we usetrueas an abbreviation forA(a)∨ ¬A(a),3φas an abbreviation for trueUφ(diamond, which should be read as “sometime in the future”), and 2φas an abbreviation for¬3¬φ(box, which should be read as “always in the future”). The semantics ofALC-LTL is based onALC-LTL structures, which are sequences ofALC-interpretations over the same non-empty domain∆ (con- stant domain assumption). We assume that every individual name stands for a unique element of ∆ (rigid individual names), and we make the unique name assumption (UNA), i.e., different individual names are interpreted by different elements of∆.

Definition 2. An ALC-LTL structure is a sequence I = (Ii)_i=0,1,... of ALC- interpretationsIi= (∆,·Îⁱ)obeying the UNA (calledworlds) such thataÎⁱ =aÎ^j for all individual namesaand alli, j∈ {0,1,2, . . .}. Given anALC-LTL formula φ, an ALC-LTL structure I = (I_i)_i=0,1,..., and a time point i ∈ {0,1,2, . . .}, validity ofφ inI at timei(written I, i|=φ) is defined inductively:

I, i|=φUψ iff there isk≥i such thatI, k|=ψ andI, j|=φ for allj, i≤j < k

As mentioned above, for some concepts and roles it is not desirable that their interpretation changes over time. Thus, we will sometimes assume that a subset of the set of concept and role names can be designated as being rigid. We will call the elements of this subsetrigid concept names andrigid role names.

Definition 3. We say that the ALC-LTL structure I = (Ii)i=0,1,... respects rigid concept names (role names) iff AÎⁱ =AÎ^j (rÎⁱ =rÎ^j) holds for all i, j ∈ {0,1,2, . . .} and all rigid concept names A(rigid role namesr).

3 The satisfiability problem in ALC-LTL

Depending on whether rigid concept and role names are considered or not, we obtain different variants of the satisfiability problem.

Definition 4. Let φbe an ALC-LTL formula and assume that a subset of the set of concept and role names has been designated as being rigid.

– We say that φ is satisfiable w.r.t. rigid names iff there is an ALC-LTL structureIrespecting rigid concept and role names such that I,0|=φ.

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W.r.t. rigid names W.r.t. rigid concepts Without rigid names ALC-LTL 2-ExpTime-complete NExpTime-complete ExpTime-complete ALC-LTL|gGCI 2-ExpTime-complete ExpTime-complete ExpTime-complete Table 1.Complexity of the satisfiability problem inALC-LTL andALC-LTL|gGCI.

– We say that φ is satisfiable w.r.t. rigid concepts iff there is an ALC-LTL structureIrespecting rigid concept names such thatI,0|=φ.

– We say that φ is satisfiable without rigid names (or simply satisfiable) iff there is anALC-LTL structureI such thatI,0|=φ.

In this paper, we show that the complexity of the satisfiability problem for ALC-LTL strongly depends on which of the above cases one considers. Note that it does not really make sense to consider satisfiability w.r.t. rigid role names, but without rigid concept names, as a separate case when investigating the complexity of the satisfiability problem. In fact, rigid concepts can be simulated by rigid roles: just introduce a new rigid role name r_A for each rigid concept nameA, and then replaceAby∃rA.>.

Another dimension that influences the complexity of the satisfiability problem is whether GCIs occur globally or locally in the formula. Intuitively, a GCI occurs globally if it must hold in every world of theALC-LTL structure.

Definition 5. We say thatφis anALC-LTL formula with global GCIsiff it is of the form φ=2B ∧ϕwhere B is a conjunction of ALC-axioms and ϕis an ALC-LTL formula that does not contain GCIs. We denote the fragment ofALC- LTL that contains onlyALC-LTL formulae with global GCIs byALC-LTL|gGCI. Note that saying, in the above definition, that B is a conjunction of ALC- axioms just means that B is a TBox together with an ABox. We could have restricted B to being a conjunction of GCIs (i.e., a TBox) since assertions α in B could be moved as conjuncts 2αto ϕ.³ However, it turns out to be more convenient to allow also ABox assertions to occur in the “global part” 2B of φ. Also note that it is important to restrict B to being a conjunction of ALC- axioms rather than an arbitrary BooleanALC-knowledge base. In fact, the lower complexity for the case of satisfiability w.r.t. rigid concepts obtained in this case (see Table 1) would not hold without this restriction (see Corollary 6.8 in [8]).

Table 1 summarizes the results of our investigation of the complexity of the satisfiability problem inALC-LTL and its fragments.

4 Reasoning with rigid names

In this section, we investigate the complexity of the satisfiability problem in ALC-LTL andALC-LTL|gGCIif rigid concepts and roles are available.

3 This is the reason why we talk aboutALC-LTL formulaewith global GCIs in this case, rather than aboutALC-LTL formulae with global axioms.

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Theorem 1. Satisfiability w.r.t. rigid names is 2-ExpTime-complete both in ALC-LTL and inALC-LTL|_gGCI.

2-ExpTime-hardness for satisfiability w.r.t. rigid names and with global GCIs (i.e., in ALC-LTL|gGCI) can be shown by a (quite intricate) reduction of the word problem for exponentially space bounded alternating Turing ma- chines (see [8]). Obviously, this also yields 2-ExpTime-hardness for the more general case with arbitrary GCIs (i.e., inALC-LTL).

In the following, we prove the complexity upper bound for ALC-LTL. Ob- viously, this also establishes the same upper bound for the restricted case of ALC-LTL|gGCI. Thus, letφbe anALC-LTL formula to be tested for satisfiability w.r.t. rigid names. We build its propositional abstractionφbby replacing each ALC-axiom by a propositional variable such that there is a 1–1 relationship between theALC-axiomsα₁, . . . , α_n occurring inφand the propositional variables p₁, . . . , p_n used for the abstraction. We assume in the following thatp_iwas used to replaceα_i (i= 1, . . . , n).

Consider a setS ⊆ P({p1, . . . , pn}), i.e., a set of subsets of{p1, . . . , pn}. Such a set induces the following (propositional) LTL formula:

φbS :=φb∧2



 _

X∈S





^

p∈X

p∧ ^

p6∈X

¬p









If φ is satisfiable in an ALC-LTL structure I = (I_i)_i=0,1,..., then there is an S ⊆ P({p1, . . . , pn}) such thatφb_S is satisfiable in a propositional LTL structure.

In fact, for eachALC-interpretationIi ofI, we define the set X_i:={pj |1≤j≤nandIi satisfiesα_j},

and then take S ={Xi |i = 0,1, . . .}. The fact that I satisfiesφ implies that its propositional abstraction satisfies φb_S, where the propositional abstraction bI = (wi)i=0,1,... of I is defined such that world wi makes variable pj true iff Ii satisfies αj. However, guessing such a set S ⊆ P({p1, . . . , pn}) and then testing whether the induced propositional LTL formula φb_S is satisfiable is not sufficient for checking satisfiability w.r.t. rigid names of the ALC-LTL formula φ. We must also check whether the guessed setScan indeed be induced by some ALC-LTL structure that respects the rigid concept and role names.

To this purpose, assume that a setS ={X₁, . . . , X_k} ⊆ P({p₁, . . . , p_n}) is given. For every i,1 ≤ i ≤k, and every flexible concept nameA (flexible role name r) occurring in α1, . . . , αn, we introduce a copy A⁽ⁱ⁾ (r⁽ⁱ⁾). We call A⁽ⁱ⁾ (r⁽ⁱ⁾) theith copy ofA(r). TheALC-axiomα⁽ⁱ⁾_j is obtained fromαjby replacing every occurrence of a flexible name by itsith copy. The setsXi(1≤i≤k) induce the following BooleanALC-knowledge bases:

Bi:= ^

pj∈Xi

α⁽ⁱ⁾_j ∧ ^

pj6∈Xi

¬α⁽ⁱ⁾_j

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Lemma 1. The ALC-LTL formula φ is satisfiable w.r.t. rigid names iff there is a set S = {X₁, . . . , X_k} ⊆ P({p₁, . . . , p_n}) such that the propositional LTL formula φb_S is satisfiable and the Boolean ALC-knowledge base B:=V

1≤i≤kB_i is consistent.

A detailed proof of this lemma can be found in [8]. It remains to show that it provides us with a decision procedure for satisfiability in ALC-LTL w.r.t. rigid names that runs in deterministic double-exponential time.

First, note that there are 2²ⁿmany subsetsS ofP({p1, . . . , pn}) to be tested, wherenis of course linearly bounded by the size ofφ. For each of these subsets S = {X1, . . . , Xk}, whose cardinality k is bounded by 2ⁿ, we need to check satisfiability ofφb_S and consistency ofB=V

1≤i≤kBi.

The size of φb_S is at most exponential in the size of φ, and the complexity of the satisfiability problem in propositional LTL is in PSpace, and thus in particular inExpTime. Consequently, satisfiability ofφb_Scan be tested in double- exponential time in the size ofφ.

The Boolean ALC-knowledge base B is a conjunction of k ≤ 2ⁿ Boolean ALC-knowledge bases Bi, where the size of each Bi is polynomial in the size of φ. The consistency problem for Boolean ALC-knowledge base is ExpTime- complete (see, e.g., Theorem 2.27 in [11]). Consequently, consistency of B can also be tested in double-exponential time in the size of the input formulaφ.

Overall, we thus have double-exponentially many tests, where each test takes double-exponential time. This provides us with a double-exponential bound for testing satisfiability inALC-LTL w.r.t. rigid names based on Lemma 1.

5 Reasoning with rigid concepts

In this section, we consider the case where rigid concept names are available, but not rigid role names. First, note that, in contrast to temporal DLs where temporal operator may occur inside of concept descriptions, rigid concept names cannot easily be expressed within the logic without rigid concept names. In fact, the GCIsAv2Aand¬Av2¬Aexpress thatAmust be interpreted in a rigid way. However, they are not allowed by the syntax ofALC-LTL since the box is applied directly to a concept, and not to an axiom. We will show below that, for ALC-LTL, the presence of rigid concept names indeed increases the complexity of the satisfiability problem, unless GCIs are restricted to being global. First, we treat the case of arbitrary GCIs, and then the special case of global GCIs.

Theorem 2. Satisfiability in ALC-LTL w.r.t. rigid concepts is NExpTime- complete.

A detailed proof of the lower bound can be found in [8]. In the proof of the upper bound, we want to reuse Lemma 1. If we apply this lemma in the case where only concept names can be rigid, then we know that the Boolean ALC-knowledge bases Bi are built over disjoint sets of role names. The only

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shared names are the rigid concept names. Obviously, we can guess a set S = {X₁, . . . , X_k} ⊆ P({p₁, . . . , p_n}), within NExpTime. However, there are two obstacles on our way to a NExpTimedecision procedure.

First, the propositional LTL formulaφb_S is of size exponential in the size of φ. Thus, a direct application of the PSpace decision procedure for satisfiability in propositional LTL would only yield an ExpSpace upper bound, which is not good enough. However, note that the only effect of the box-formula in φb

Sb is to restrict the worldsw in a propositional LTL structure satisfying φb to being induced by one of the elements of S. One way of deciding satisfiabilityb of a propositional LTL formula φb is to construct a B¨uchi automatonA

φbthat accepts the propositional LTL structures satisfying φ. To be more precise, letb Σ:=P({p₁, . . . , p_n}). Then a given propositional LTL structurebI= (w_ι)_ι=0,1,...

can be represented by an infinite wordX₀X₁. . .overΣ, whereX_ιconsists of the propositional variables that wι makes true. The B¨uchi automaton A

φb is built such that it accepts exactly those infinite words overΣ that represent propositional LTL structures satisfyingφ. Consequently,b φbis satisfiable iff the language accepted byA

φbis non-empty. The size ofA

φbis exponential in the size ofφ, andb the emptiness test for B¨uchi automata is polynomial in the size of the automaton. The automatonA

φbcan now easily be modified into one accepting exactly the words representing propositional LTL structures satisfying φb

Sb. In fact, we just need to remove all transitions that use a letter fromΣ\S. Obviously, thisb modification can be done in time polynomial in the size ofA

φb, and thus in time exponential in the size ofφ. The size of the resulting automaton is obviously stillb only exponential in the size of φ, and thus its emptiness can be tested in timeb exponential in the size ofφb(and hence ofφ).

The second obstacle is the fact that B = V

1≤i≤kBi is of exponential size, and thus testing it directly for consistency using the known ExpTimedecision procedure for satisfiability of Boolean ALC-knowledge bases would provide us with a double-exponential time bound. Instead of testing the consistency of B directly we reduce this test tok separate consistency tests, each requiring time exponential in the size of φ. Before we can do this, we need another guessing step. Assume that A1, . . . , Ar are all the rigid concept names occurring in φ, and thata1, . . . , as are all the individual names occurring in φ. We guess a set T ⊆ P({A1, . . . , Ar}) and a mappingt:{a1, . . . , as} → T. Again, this guess can clearly be done withinNExpTime.

GivenT andt, we extend the knowledge basesBito knowledge basesBbi(T,t) as follows. For Y ⊆ {A1, . . . , Ar}, letCY be the concept description

CY :=_A∈Y

u

^A^u_A6∈Y

u

^¬A.

We defineBbi(T,t) as Bbi(T,t) :=Bi∧ ^

t(a)=Y

a:C_Y ∧ > v_Y

t

∈TC_Y ∧ ^

Y∈T

¬(> v ¬CY).

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Lemma 2. The Boolean ALC-knowledge base B:=V

1≤i≤kB_i is consistent iff there is a setT ⊆ P({A₁, . . . , A_r})and a mappingt:{a₁, . . . , a_s} → T such that the Boolean knowledge bases Bbi(T,t)fori= 1, . . . , kare separately consistent.

A proof of this lemma can be found in [8]. To finish the proof of Theorem 2, we must show that the consistency of Bbi(T,t) can be decided in time exponential in the size of the input formulaφ. Note that this is not trivial. In fact, while the size ofBi ∧ V

t(a)=Y a:CY is polynomial in the size of φ, the cardinality of

T, and thus the size of

> v_Y

t

_∈T^C^Y ^∧ ^{^}

Y∈T

¬(> v ¬C_Y), (3)

can be exponential in the size of φ. Decidability of the consistency of Bb_i(T,t) in time exponential in the size of φ is, however, an immediate consequence of the next lemma. To formulate this lemma, we need to introduce one more no- tation. Let Bbbe a BooleanALC-knowledge base of size n, A1, . . . , Ar concept names occurring in B, andb T ⊆ P({A1, . . . , Ar}). Note that this implies that the cardinality of T is at most exponential in n, and the size of eachY ∈ T is linear inn. We say thatBbis consistent w.r.t. T iff it has a model that is also a model of (3). The following lemma can be shown by an adaptation of the proof of Theorem 2.27 in [11], which shows that the consistency problem for Boolean ALC-knowledge bases is inExpTime(see [8] for details).

Lemma 3. Let Bbbe a BooleanALC-knowledge base of size n,A1, . . . , Ar concept names occurring in B, andb T ⊆ P({A1, . . . , Ar}). Then, consistency of Bb w.r.t.T can be decided in time exponential inn.

Overall, this completes the proof of Theorem 2. In fact, after twoNExpTime guesses, all we have to do arek(i.e., exponentially many)ExpTimeconsistency tests.

Restricting GCIs to global ones decreases the complexity of the satisfiability problem.

Theorem 3. Satisfiability inALC-LTL|gGCI w.r.t. rigid concepts isExpTime- complete.

ExpTime-hardness is an easy consequence of the well-known fact that concept satisfiability inALC w.r.t. a single GCI is ExpTime-complete:C is satisfiable w.r.t.D₁vD₂ iffa:C∧2(D₁vD₂) is satisfiable.

To prove theExpTimeupper bound, we consider anALC-LTL formulaφ= 2B ∧ϕ, whereBis a conjunction ofALC-axioms andϕis anALC-LTL formula that does not contain GCIs. We say thatBisφ-exhaustiveif, for every individual name a and every rigid concept name A, either a : A or a : ¬A occurs as a conjunct inB. We can assume without loss of generality thatBisφ-exhaustive.

In fact, given an arbitrary BooleanALC-knowledge baseB, we can build all the

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φ-exhaustive knowledge basesB⁰that are obtained fromBby conjoining to it, for every individual nameaand every rigid concept nameA, eithera:Aora:¬A.

Obviously,φ=2B ∧ϕis satisfiable w.r.t. rigid concepts iff2B⁰∧ϕis satisfiable w.r.t. rigid concepts for one of the extension B⁰ of B obtained this way. Since the size of each such an extension is polynomial and there are only exponentially many such extensions, it is sufficient to show that testing satisfiability of2B⁰∧ϕ w.r.t. rigid concepts for φ-exhaustive knowledge basesB⁰ is inExpTime.

Following the approach used in the proof of Theorem 1, we abstract every ABox assertion αi occurring in ϕ by a propositional variable pi, thus building the propositional LTL-formulaϕ. Next, we compute the setb S, which consists ofb thoseX ⊆ {p1, . . . , pn} for which the BooleanALC-knowledge base

BX := B ∧ ^

pj∈X

αj∧ ^

pj6∈X

¬αj

is consistent. This computation can be done in exponential time since it requires exponentially manyExpTime consistency tests.

Lemma 4. Let φ = 2B ∧ϕ be such that B is a φ-exhaustive conjunction of ALC-axioms and ϕ is an ALC-LTL formula not containing GCIs. Then φ is satisfiable w.r.t. rigid concepts iff the propositional LTL formula

ϕb

Sb := ϕb∧2(_

X∈Sb

( ^

p_j∈X

pj∧ ^

p_j6∈X

¬pj))

is satisfiable.

The proof of this lemma can again be found in [8]. Note that this actually completes the proof of Theorem 3. In fact, as shown in the proof of Theorem 2, satisfiability ofϕb

Sbcan be decided in exponential time.

6 Reasoning without rigid names

In this section, we consider the case where we have no rigid names at all.

Theorem 4. Satisfiability without rigid names inALC-LTL and in its fragment ALC-LTL|gGCI isExpTime-complete.

ExpTime-hardness is again an easy consequence of the fact that concept satisfiability inALCw.r.t. a single GCI isExpTime-complete. As already mentioned in the introduction, theExpTimeupper bound follows from more general results proved in Chapter 11 of [11] (see the remark following Theorem 14.14 on page 605 of [11]). A direct proof of the upper bound, which is similar to the proof of Theorem 3, is given in [8].

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